3.2.38 \(\int (c e+d e x)^3 (a+b \sinh ^{-1}(c+d x))^3 \, dx\) [138]

Optimal. Leaf size=279 \[ \frac {45 b^3 e^3 (c+d x) \sqrt {1+(c+d x)^2}}{256 d}-\frac {3 b^3 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{128 d}-\frac {45 b^3 e^3 \sinh ^{-1}(c+d x)}{256 d}-\frac {9 b^2 e^3 (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )}{32 d}+\frac {3 b^2 e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )}{32 d}+\frac {9 b e^3 (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{32 d}-\frac {3 b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{16 d}-\frac {3 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{4 d} \]

[Out]

-45/256*b^3*e^3*arcsinh(d*x+c)/d-9/32*b^2*e^3*(d*x+c)^2*(a+b*arcsinh(d*x+c))/d+3/32*b^2*e^3*(d*x+c)^4*(a+b*arc
sinh(d*x+c))/d-3/32*e^3*(a+b*arcsinh(d*x+c))^3/d+1/4*e^3*(d*x+c)^4*(a+b*arcsinh(d*x+c))^3/d+45/256*b^3*e^3*(d*
x+c)*(1+(d*x+c)^2)^(1/2)/d-3/128*b^3*e^3*(d*x+c)^3*(1+(d*x+c)^2)^(1/2)/d+9/32*b*e^3*(d*x+c)*(a+b*arcsinh(d*x+c
))^2*(1+(d*x+c)^2)^(1/2)/d-3/16*b*e^3*(d*x+c)^3*(a+b*arcsinh(d*x+c))^2*(1+(d*x+c)^2)^(1/2)/d

________________________________________________________________________________________

Rubi [A]
time = 0.26, antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {5859, 12, 5776, 5812, 5783, 327, 221} \begin {gather*} \frac {3 b^2 e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )}{32 d}-\frac {9 b^2 e^3 (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{4 d}-\frac {3 b e^3 \sqrt {(c+d x)^2+1} (c+d x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^2}{16 d}+\frac {9 b e^3 \sqrt {(c+d x)^2+1} (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^2}{32 d}-\frac {3 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{32 d}-\frac {3 b^3 e^3 \sqrt {(c+d x)^2+1} (c+d x)^3}{128 d}+\frac {45 b^3 e^3 \sqrt {(c+d x)^2+1} (c+d x)}{256 d}-\frac {45 b^3 e^3 \sinh ^{-1}(c+d x)}{256 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^3*(a + b*ArcSinh[c + d*x])^3,x]

[Out]

(45*b^3*e^3*(c + d*x)*Sqrt[1 + (c + d*x)^2])/(256*d) - (3*b^3*e^3*(c + d*x)^3*Sqrt[1 + (c + d*x)^2])/(128*d) -
(45*b^3*e^3*ArcSinh[c + d*x])/(256*d) - (9*b^2*e^3*(c + d*x)^2*(a + b*ArcSinh[c + d*x]))/(32*d) + (3*b^2*e^3*
(c + d*x)^4*(a + b*ArcSinh[c + d*x]))/(32*d) + (9*b*e^3*(c + d*x)*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x
])^2)/(32*d) - (3*b*e^3*(c + d*x)^3*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^2)/(16*d) - (3*e^3*(a + b*A
rcSinh[c + d*x])^3)/(32*d) + (e^3*(c + d*x)^4*(a + b*ArcSinh[c + d*x])^3)/(4*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 5776

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcS
inh[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[
1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5783

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*S
imp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ
[e, c^2*d] && NeQ[n, -1]

Rule 5812

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[
f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(e*(m + 2*p + 1))), x] + (-Dist[f^2*((m - 1)/(c^2*
(m + 2*p + 1))), Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Dist[b*f*(n/(c*(m + 2*p + 1)
))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p], Int[(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1)
, x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0
]

Rule 5859

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[
Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x
]

Rubi steps

\begin {align*} \int (c e+d e x)^3 \left (a+b \sinh ^{-1}(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int e^3 x^3 \left (a+b \sinh ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 \text {Subst}\left (\int x^3 \left (a+b \sinh ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{4 d}-\frac {\left (3 b e^3\right ) \text {Subst}\left (\int \frac {x^4 \left (a+b \sinh ^{-1}(x)\right )^2}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{4 d}\\ &=-\frac {3 b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{16 d}+\frac {e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{4 d}+\frac {\left (9 b e^3\right ) \text {Subst}\left (\int \frac {x^2 \left (a+b \sinh ^{-1}(x)\right )^2}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{16 d}+\frac {\left (3 b^2 e^3\right ) \text {Subst}\left (\int x^3 \left (a+b \sinh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{8 d}\\ &=\frac {3 b^2 e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )}{32 d}+\frac {9 b e^3 (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{32 d}-\frac {3 b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{16 d}+\frac {e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{4 d}-\frac {\left (9 b e^3\right ) \text {Subst}\left (\int \frac {\left (a+b \sinh ^{-1}(x)\right )^2}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{32 d}-\frac {\left (9 b^2 e^3\right ) \text {Subst}\left (\int x \left (a+b \sinh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{16 d}-\frac {\left (3 b^3 e^3\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{32 d}\\ &=-\frac {3 b^3 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{128 d}-\frac {9 b^2 e^3 (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )}{32 d}+\frac {3 b^2 e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )}{32 d}+\frac {9 b e^3 (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{32 d}-\frac {3 b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{16 d}-\frac {3 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{4 d}+\frac {\left (9 b^3 e^3\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{128 d}+\frac {\left (9 b^3 e^3\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{32 d}\\ &=\frac {45 b^3 e^3 (c+d x) \sqrt {1+(c+d x)^2}}{256 d}-\frac {3 b^3 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{128 d}-\frac {9 b^2 e^3 (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )}{32 d}+\frac {3 b^2 e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )}{32 d}+\frac {9 b e^3 (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{32 d}-\frac {3 b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{16 d}-\frac {3 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{4 d}-\frac {\left (9 b^3 e^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{256 d}-\frac {\left (9 b^3 e^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{64 d}\\ &=\frac {45 b^3 e^3 (c+d x) \sqrt {1+(c+d x)^2}}{256 d}-\frac {3 b^3 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2}}{128 d}-\frac {45 b^3 e^3 \sinh ^{-1}(c+d x)}{256 d}-\frac {9 b^2 e^3 (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )}{32 d}+\frac {3 b^2 e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )}{32 d}+\frac {9 b e^3 (c+d x) \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{32 d}-\frac {3 b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{16 d}-\frac {3 e^3 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{4 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.24, size = 303, normalized size = 1.09 \begin {gather*} \frac {e^3 \left (-72 a b^2 (c+d x)^2+8 a \left (8 a^2+3 b^2\right ) (c+d x)^4+3 b (c+d x) \sqrt {1+(c+d x)^2} \left (3 \left (8 a^2+5 b^2\right )-2 \left (8 a^2+b^2\right ) (c+d x)^2\right )-9 b \left (8 a^2+5 b^2\right ) \sinh ^{-1}(c+d x)-24 b (c+d x) \left (3 b^2 (c+d x)-8 a^2 (c+d x)^3-b^2 (c+d x)^3-6 a b \sqrt {1+(c+d x)^2}+4 a b (c+d x)^2 \sqrt {1+(c+d x)^2}\right ) \sinh ^{-1}(c+d x)+24 b^2 \left (-3 a+8 a (c+d x)^4+3 b (c+d x) \sqrt {1+(c+d x)^2}-2 b (c+d x)^3 \sqrt {1+(c+d x)^2}\right ) \sinh ^{-1}(c+d x)^2+8 b^3 \left (-3+8 (c+d x)^4\right ) \sinh ^{-1}(c+d x)^3\right )}{256 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)^3*(a + b*ArcSinh[c + d*x])^3,x]

[Out]

(e^3*(-72*a*b^2*(c + d*x)^2 + 8*a*(8*a^2 + 3*b^2)*(c + d*x)^4 + 3*b*(c + d*x)*Sqrt[1 + (c + d*x)^2]*(3*(8*a^2
+ 5*b^2) - 2*(8*a^2 + b^2)*(c + d*x)^2) - 9*b*(8*a^2 + 5*b^2)*ArcSinh[c + d*x] - 24*b*(c + d*x)*(3*b^2*(c + d*
x) - 8*a^2*(c + d*x)^3 - b^2*(c + d*x)^3 - 6*a*b*Sqrt[1 + (c + d*x)^2] + 4*a*b*(c + d*x)^2*Sqrt[1 + (c + d*x)^
2])*ArcSinh[c + d*x] + 24*b^2*(-3*a + 8*a*(c + d*x)^4 + 3*b*(c + d*x)*Sqrt[1 + (c + d*x)^2] - 2*b*(c + d*x)^3*
Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x]^2 + 8*b^3*(-3 + 8*(c + d*x)^4)*ArcSinh[c + d*x]^3))/(256*d)

________________________________________________________________________________________

Maple [A]
time = 3.96, size = 409, normalized size = 1.47

method	result	size
default	\(\frac {e^{3} \left (d x +c \right )^{4} a^{3}}{4 d}-\frac {b^{3} e^{3} \left (-32 \left (\cosh ^{2}\left (2 \arcsinh \left (d x +c \right )\right )\right ) \arcsinh \left (d x +c \right )^{3}+24 \sinh \left (2 \arcsinh \left (d x +c \right )\right ) \cosh \left (2 \arcsinh \left (d x +c \right )\right ) \arcsinh \left (d x +c \right )^{2}+64 \cosh \left (2 \arcsinh \left (d x +c \right )\right ) \arcsinh \left (d x +c \right )^{3}-96 \arcsinh \left (d x +c \right )^{2} \sinh \left (2 \arcsinh \left (d x +c \right )\right )-12 \arcsinh \left (d x +c \right ) \left (\cosh ^{2}\left (2 \arcsinh \left (d x +c \right )\right )\right )+16 \arcsinh \left (d x +c \right )^{3}+3 \sinh \left (2 \arcsinh \left (d x +c \right )\right ) \cosh \left (2 \arcsinh \left (d x +c \right )\right )+96 \cosh \left (2 \arcsinh \left (d x +c \right )\right ) \arcsinh \left (d x +c \right )-48 \sinh \left (2 \arcsinh \left (d x +c \right )\right )+6 \arcsinh \left (d x +c \right )\right )}{512 d}-\frac {3 a \,b^{2} e^{3} \left (-16 \left (\cosh ^{2}\left (2 \arcsinh \left (d x +c \right )\right )\right ) \arcsinh \left (d x +c \right )^{2}+8 \sinh \left (2 \arcsinh \left (d x +c \right )\right ) \cosh \left (2 \arcsinh \left (d x +c \right )\right ) \arcsinh \left (d x +c \right )+32 \cosh \left (2 \arcsinh \left (d x +c \right )\right ) \arcsinh \left (d x +c \right )^{2}-32 \arcsinh \left (d x +c \right ) \sinh \left (2 \arcsinh \left (d x +c \right )\right )-2 \left (\cosh ^{2}\left (2 \arcsinh \left (d x +c \right )\right )\right )+8 \arcsinh \left (d x +c \right )^{2}+16 \cosh \left (2 \arcsinh \left (d x +c \right )\right )+1\right )}{256 d}+\frac {3 a^{2} b \,e^{3} \left (\frac {\left (d x +c \right )^{4} \arcsinh \left (d x +c \right )}{4}-\frac {\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{16}+\frac {3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{32}-\frac {3 \arcsinh \left (d x +c \right )}{32}\right )}{d}\)	\(409\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^3*(a+b*arcsinh(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/4*e^3*(d*x+c)^4*a^3/d-1/512*b^3*e^3*(-32*cosh(2*arcsinh(d*x+c))^2*arcsinh(d*x+c)^3+24*sinh(2*arcsinh(d*x+c))
*cosh(2*arcsinh(d*x+c))*arcsinh(d*x+c)^2+64*cosh(2*arcsinh(d*x+c))*arcsinh(d*x+c)^3-96*arcsinh(d*x+c)^2*sinh(2
*arcsinh(d*x+c))-12*arcsinh(d*x+c)*cosh(2*arcsinh(d*x+c))^2+16*arcsinh(d*x+c)^3+3*sinh(2*arcsinh(d*x+c))*cosh(
2*arcsinh(d*x+c))+96*cosh(2*arcsinh(d*x+c))*arcsinh(d*x+c)-48*sinh(2*arcsinh(d*x+c))+6*arcsinh(d*x+c))/d-3/256
*a*b^2*e^3*(-16*cosh(2*arcsinh(d*x+c))^2*arcsinh(d*x+c)^2+8*sinh(2*arcsinh(d*x+c))*cosh(2*arcsinh(d*x+c))*arcs
inh(d*x+c)+32*cosh(2*arcsinh(d*x+c))*arcsinh(d*x+c)^2-32*arcsinh(d*x+c)*sinh(2*arcsinh(d*x+c))-2*cosh(2*arcsin
h(d*x+c))^2+8*arcsinh(d*x+c)^2+16*cosh(2*arcsinh(d*x+c))+1)/d+3*a^2*b*e^3/d*(1/4*(d*x+c)^4*arcsinh(d*x+c)-1/16
*(d*x+c)^3*(1+(d*x+c)^2)^(1/2)+3/32*(d*x+c)*(1+(d*x+c)^2)^(1/2)-3/32*arcsinh(d*x+c))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^3*(a+b*arcsinh(d*x+c))^3,x, algorithm="maxima")

[Out]

1/4*a^3*d^3*x^4*e^3 + a^3*c*d^2*x^3*e^3 + 3/2*a^3*c^2*d*x^2*e^3 + 9/4*(2*x^2*arcsinh(d*x + c) - d*(3*c^2*arcsi
nh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^3 + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*x/d^2 - (c^2 +
1)*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^3 - 3*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c/d^3
))*a^2*b*c^2*d*e^3 + 1/2*(6*x^3*arcsinh(d*x + c) - d*(2*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*x^2/d^2 - 15*c^3*arc
sinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^4 - 5*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c*x/d^3 + 9
*(c^2 + 1)*c*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^4 + 15*sqrt(d^2*x^2 + 2*c*d*x + c^2
+ 1)*c^2/d^4 - 4*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(c^2 + 1)/d^4))*a^2*b*c*d^2*e^3 + 1/32*(24*x^4*arcsinh(d*x
+ c) - (6*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*x^3/d^2 - 14*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c*x^2/d^3 + 105*c^
4*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^5 + 35*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c^2*x
/d^4 - 90*(c^2 + 1)*c^2*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^5 - 105*sqrt(d^2*x^2 + 2
*c*d*x + c^2 + 1)*c^3/d^5 - 9*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(c^2 + 1)*x/d^4 + 9*(c^2 + 1)^2*arcsinh(2*(d^2
*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^5 + 55*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(c^2 + 1)*c/d^5)*d)*a
^2*b*d^3*e^3 + a^3*c^3*x*e^3 + 3*((d*x + c)*arcsinh(d*x + c) - sqrt((d*x + c)^2 + 1))*a^2*b*c^3*e^3/d + 1/4*(b
^3*d^3*x^4*e^3 + 4*b^3*c*d^2*x^3*e^3 + 6*b^3*c^2*d*x^2*e^3 + 4*b^3*c^3*x*e^3)*log(d*x + c + sqrt(d^2*x^2 + 2*c
*d*x + c^2 + 1))^3 + integrate(3/4*((4*a*b^2*d^6 - b^3*d^6)*x^6*e^3 + 6*(4*a*b^2*c*d^5 - b^3*c*d^5)*x^5*e^3 +
(4*(15*c^2*d^4 + d^4)*a*b^2 - (15*c^2*d^4 + d^4)*b^3)*x^4*e^3 + 4*(c^6 + c^4)*a*b^2*e^3 + 4*(4*(5*c^3*d^3 + c*
d^3)*a*b^2 - (5*c^3*d^3 + c*d^3)*b^3)*x^3*e^3 + 2*(6*(5*c^4*d^2 + 2*c^2*d^2)*a*b^2 - (7*c^4*d^2 + 3*c^2*d^2)*b
^3)*x^2*e^3 + 4*(2*(3*c^5*d + 2*c^3*d)*a*b^2 - (c^5*d + c^3*d)*b^3)*x*e^3 + ((4*a*b^2*d^5 - b^3*d^5)*x^5*e^3 +
5*(4*a*b^2*c*d^4 - b^3*c*d^4)*x^4*e^3 + 4*(c^5 + c^3)*a*b^2*e^3 - 2*(5*b^3*c^2*d^3 - 2*(10*c^2*d^3 + d^3)*a*b
^2)*x^3*e^3 - 2*(5*b^3*c^3*d^2 - 2*(10*c^3*d^2 + 3*c*d^2)*a*b^2)*x^2*e^3 - 4*(b^3*c^4*d - (5*c^4*d + 3*c^2*d)*
a*b^2)*x*e^3)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^2/(d^3*x^3 +
3*c*d^2*x^2 + c^3 + (3*c^2*d + d)*x + (d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + c), x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2632 vs. \(2 (244) = 488\).
time = 0.45, size = 2632, normalized size = 9.43 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^3*(a+b*arcsinh(d*x+c))^3,x, algorithm="fricas")

[Out]

1/256*(8*((8*a^3 + 3*a*b^2)*d^4*x^4 + 4*(8*a^3 + 3*a*b^2)*c*d^3*x^3 - 3*(3*a*b^2 - 2*(8*a^3 + 3*a*b^2)*c^2)*d^
2*x^2 - 2*(9*a*b^2*c - 2*(8*a^3 + 3*a*b^2)*c^3)*d*x)*cosh(1)^3 + 8*((8*b^3*d^4*x^4 + 32*b^3*c*d^3*x^3 + 48*b^3
*c^2*d^2*x^2 + 32*b^3*c^3*d*x + 8*b^3*c^4 - 3*b^3)*cosh(1)^3 + 3*(8*b^3*d^4*x^4 + 32*b^3*c*d^3*x^3 + 48*b^3*c^
2*d^2*x^2 + 32*b^3*c^3*d*x + 8*b^3*c^4 - 3*b^3)*cosh(1)^2*sinh(1) + 3*(8*b^3*d^4*x^4 + 32*b^3*c*d^3*x^3 + 48*b
^3*c^2*d^2*x^2 + 32*b^3*c^3*d*x + 8*b^3*c^4 - 3*b^3)*cosh(1)*sinh(1)^2 + (8*b^3*d^4*x^4 + 32*b^3*c*d^3*x^3 + 4
8*b^3*c^2*d^2*x^2 + 32*b^3*c^3*d*x + 8*b^3*c^4 - 3*b^3)*sinh(1)^3)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2
+ 1))^3 + 24*((8*a^3 + 3*a*b^2)*d^4*x^4 + 4*(8*a^3 + 3*a*b^2)*c*d^3*x^3 - 3*(3*a*b^2 - 2*(8*a^3 + 3*a*b^2)*c^2
)*d^2*x^2 - 2*(9*a*b^2*c - 2*(8*a^3 + 3*a*b^2)*c^3)*d*x)*cosh(1)^2*sinh(1) + 24*((8*a^3 + 3*a*b^2)*d^4*x^4 + 4
*(8*a^3 + 3*a*b^2)*c*d^3*x^3 - 3*(3*a*b^2 - 2*(8*a^3 + 3*a*b^2)*c^2)*d^2*x^2 - 2*(9*a*b^2*c - 2*(8*a^3 + 3*a*b
^2)*c^3)*d*x)*cosh(1)*sinh(1)^2 + 8*((8*a^3 + 3*a*b^2)*d^4*x^4 + 4*(8*a^3 + 3*a*b^2)*c*d^3*x^3 - 3*(3*a*b^2 -
2*(8*a^3 + 3*a*b^2)*c^2)*d^2*x^2 - 2*(9*a*b^2*c - 2*(8*a^3 + 3*a*b^2)*c^3)*d*x)*sinh(1)^3 + 24*((8*a*b^2*d^4*x
^4 + 32*a*b^2*c*d^3*x^3 + 48*a*b^2*c^2*d^2*x^2 + 32*a*b^2*c^3*d*x + 8*a*b^2*c^4 - 3*a*b^2)*cosh(1)^3 + 3*(8*a*
b^2*d^4*x^4 + 32*a*b^2*c*d^3*x^3 + 48*a*b^2*c^2*d^2*x^2 + 32*a*b^2*c^3*d*x + 8*a*b^2*c^4 - 3*a*b^2)*cosh(1)^2*
sinh(1) + 3*(8*a*b^2*d^4*x^4 + 32*a*b^2*c*d^3*x^3 + 48*a*b^2*c^2*d^2*x^2 + 32*a*b^2*c^3*d*x + 8*a*b^2*c^4 - 3*
a*b^2)*cosh(1)*sinh(1)^2 + (8*a*b^2*d^4*x^4 + 32*a*b^2*c*d^3*x^3 + 48*a*b^2*c^2*d^2*x^2 + 32*a*b^2*c^3*d*x + 8
*a*b^2*c^4 - 3*a*b^2)*sinh(1)^3 - sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*((2*b^3*d^3*x^3 + 6*b^3*c*d^2*x^2 + 2*b^3*
c^3 - 3*b^3*c + 3*(2*b^3*c^2 - b^3)*d*x)*cosh(1)^3 + 3*(2*b^3*d^3*x^3 + 6*b^3*c*d^2*x^2 + 2*b^3*c^3 - 3*b^3*c
+ 3*(2*b^3*c^2 - b^3)*d*x)*cosh(1)^2*sinh(1) + 3*(2*b^3*d^3*x^3 + 6*b^3*c*d^2*x^2 + 2*b^3*c^3 - 3*b^3*c + 3*(2
*b^3*c^2 - b^3)*d*x)*cosh(1)*sinh(1)^2 + (2*b^3*d^3*x^3 + 6*b^3*c*d^2*x^2 + 2*b^3*c^3 - 3*b^3*c + 3*(2*b^3*c^2
- b^3)*d*x)*sinh(1)^3))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^2 + 3*((8*(8*a^2*b + b^3)*d^4*x^4 +
32*(8*a^2*b + b^3)*c*d^3*x^3 - 24*b^3*c^2 + 8*(8*a^2*b + b^3)*c^4 - 24*(b^3 - 2*(8*a^2*b + b^3)*c^2)*d^2*x^2 -
24*a^2*b - 15*b^3 - 16*(3*b^3*c - 2*(8*a^2*b + b^3)*c^3)*d*x)*cosh(1)^3 + 3*(8*(8*a^2*b + b^3)*d^4*x^4 + 32*(
8*a^2*b + b^3)*c*d^3*x^3 - 24*b^3*c^2 + 8*(8*a^2*b + b^3)*c^4 - 24*(b^3 - 2*(8*a^2*b + b^3)*c^2)*d^2*x^2 - 24*
a^2*b - 15*b^3 - 16*(3*b^3*c - 2*(8*a^2*b + b^3)*c^3)*d*x)*cosh(1)^2*sinh(1) + 3*(8*(8*a^2*b + b^3)*d^4*x^4 +
32*(8*a^2*b + b^3)*c*d^3*x^3 - 24*b^3*c^2 + 8*(8*a^2*b + b^3)*c^4 - 24*(b^3 - 2*(8*a^2*b + b^3)*c^2)*d^2*x^2 -
24*a^2*b - 15*b^3 - 16*(3*b^3*c - 2*(8*a^2*b + b^3)*c^3)*d*x)*cosh(1)*sinh(1)^2 + (8*(8*a^2*b + b^3)*d^4*x^4
+ 32*(8*a^2*b + b^3)*c*d^3*x^3 - 24*b^3*c^2 + 8*(8*a^2*b + b^3)*c^4 - 24*(b^3 - 2*(8*a^2*b + b^3)*c^2)*d^2*x^2
- 24*a^2*b - 15*b^3 - 16*(3*b^3*c - 2*(8*a^2*b + b^3)*c^3)*d*x)*sinh(1)^3 - 16*sqrt(d^2*x^2 + 2*c*d*x + c^2 +
1)*((2*a*b^2*d^3*x^3 + 6*a*b^2*c*d^2*x^2 + 2*a*b^2*c^3 - 3*a*b^2*c + 3*(2*a*b^2*c^2 - a*b^2)*d*x)*cosh(1)^3 +
3*(2*a*b^2*d^3*x^3 + 6*a*b^2*c*d^2*x^2 + 2*a*b^2*c^3 - 3*a*b^2*c + 3*(2*a*b^2*c^2 - a*b^2)*d*x)*cosh(1)^2*sin
h(1) + 3*(2*a*b^2*d^3*x^3 + 6*a*b^2*c*d^2*x^2 + 2*a*b^2*c^3 - 3*a*b^2*c + 3*(2*a*b^2*c^2 - a*b^2)*d*x)*cosh(1)
*sinh(1)^2 + (2*a*b^2*d^3*x^3 + 6*a*b^2*c*d^2*x^2 + 2*a*b^2*c^3 - 3*a*b^2*c + 3*(2*a*b^2*c^2 - a*b^2)*d*x)*sin
h(1)^3))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)) - 3*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*((2*(8*a^2*b +
b^3)*d^3*x^3 + 6*(8*a^2*b + b^3)*c*d^2*x^2 + 2*(8*a^2*b + b^3)*c^3 - 3*(8*a^2*b + 5*b^3 - 2*(8*a^2*b + b^3)*c
^2)*d*x - 3*(8*a^2*b + 5*b^3)*c)*cosh(1)^3 + 3*(2*(8*a^2*b + b^3)*d^3*x^3 + 6*(8*a^2*b + b^3)*c*d^2*x^2 + 2*(8
*a^2*b + b^3)*c^3 - 3*(8*a^2*b + 5*b^3 - 2*(8*a^2*b + b^3)*c^2)*d*x - 3*(8*a^2*b + 5*b^3)*c)*cosh(1)^2*sinh(1)
+ 3*(2*(8*a^2*b + b^3)*d^3*x^3 + 6*(8*a^2*b + b^3)*c*d^2*x^2 + 2*(8*a^2*b + b^3)*c^3 - 3*(8*a^2*b + 5*b^3 - 2
*(8*a^2*b + b^3)*c^2)*d*x - 3*(8*a^2*b + 5*b^3)*c)*cosh(1)*sinh(1)^2 + (2*(8*a^2*b + b^3)*d^3*x^3 + 6*(8*a^2*b
+ b^3)*c*d^2*x^2 + 2*(8*a^2*b + b^3)*c^3 - 3*(8*a^2*b + 5*b^3 - 2*(8*a^2*b + b^3)*c^2)*d*x - 3*(8*a^2*b + 5*b
^3)*c)*sinh(1)^3))/d

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1828 vs. \(2 (260) = 520\).
time = 0.90, size = 1828, normalized size = 6.55 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**3*(a+b*asinh(d*x+c))**3,x)

[Out]

Piecewise((a**3*c**3*e**3*x + 3*a**3*c**2*d*e**3*x**2/2 + a**3*c*d**2*e**3*x**3 + a**3*d**3*e**3*x**4/4 + 3*a*
*2*b*c**4*e**3*asinh(c + d*x)/(4*d) + 3*a**2*b*c**3*e**3*x*asinh(c + d*x) - 3*a**2*b*c**3*e**3*sqrt(c**2 + 2*c
*d*x + d**2*x**2 + 1)/(16*d) + 9*a**2*b*c**2*d*e**3*x**2*asinh(c + d*x)/2 - 9*a**2*b*c**2*e**3*x*sqrt(c**2 + 2
*c*d*x + d**2*x**2 + 1)/16 + 3*a**2*b*c*d**2*e**3*x**3*asinh(c + d*x) - 9*a**2*b*c*d*e**3*x**2*sqrt(c**2 + 2*c
*d*x + d**2*x**2 + 1)/16 + 9*a**2*b*c*e**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(32*d) + 3*a**2*b*d**3*e**3*x*
*4*asinh(c + d*x)/4 - 3*a**2*b*d**2*e**3*x**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/16 + 9*a**2*b*e**3*x*sqrt(c
**2 + 2*c*d*x + d**2*x**2 + 1)/32 - 9*a**2*b*e**3*asinh(c + d*x)/(32*d) + 3*a*b**2*c**4*e**3*asinh(c + d*x)**2
/(4*d) + 3*a*b**2*c**3*e**3*x*asinh(c + d*x)**2 + 3*a*b**2*c**3*e**3*x/8 - 3*a*b**2*c**3*e**3*sqrt(c**2 + 2*c*
d*x + d**2*x**2 + 1)*asinh(c + d*x)/(8*d) + 9*a*b**2*c**2*d*e**3*x**2*asinh(c + d*x)**2/2 + 9*a*b**2*c**2*d*e*
*3*x**2/16 - 9*a*b**2*c**2*e**3*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/8 + 3*a*b**2*c*d**2*e**3
*x**3*asinh(c + d*x)**2 + 3*a*b**2*c*d**2*e**3*x**3/8 - 9*a*b**2*c*d*e**3*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2
+ 1)*asinh(c + d*x)/8 - 9*a*b**2*c*e**3*x/16 + 9*a*b**2*c*e**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c +
d*x)/(16*d) + 3*a*b**2*d**3*e**3*x**4*asinh(c + d*x)**2/4 + 3*a*b**2*d**3*e**3*x**4/32 - 3*a*b**2*d**2*e**3*x
**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/8 - 9*a*b**2*d*e**3*x**2/32 + 9*a*b**2*e**3*x*sqrt(c**
2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/16 - 9*a*b**2*e**3*asinh(c + d*x)**2/(32*d) + b**3*c**4*e**3*asinh
(c + d*x)**3/(4*d) + 3*b**3*c**4*e**3*asinh(c + d*x)/(32*d) + b**3*c**3*e**3*x*asinh(c + d*x)**3 + 3*b**3*c**3
*e**3*x*asinh(c + d*x)/8 - 3*b**3*c**3*e**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**2/(16*d) - 3*
b**3*c**3*e**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(128*d) + 3*b**3*c**2*d*e**3*x**2*asinh(c + d*x)**3/2 + 9*
b**3*c**2*d*e**3*x**2*asinh(c + d*x)/16 - 9*b**3*c**2*e**3*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*
x)**2/16 - 9*b**3*c**2*e**3*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/128 - 9*b**3*c**2*e**3*asinh(c + d*x)/(32*d
) + b**3*c*d**2*e**3*x**3*asinh(c + d*x)**3 + 3*b**3*c*d**2*e**3*x**3*asinh(c + d*x)/8 - 9*b**3*c*d*e**3*x**2*
sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**2/16 - 9*b**3*c*d*e**3*x**2*sqrt(c**2 + 2*c*d*x + d**2*x*
*2 + 1)/128 - 9*b**3*c*e**3*x*asinh(c + d*x)/16 + 9*b**3*c*e**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c +
d*x)**2/(32*d) + 45*b**3*c*e**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(256*d) + b**3*d**3*e**3*x**4*asinh(c +
d*x)**3/4 + 3*b**3*d**3*e**3*x**4*asinh(c + d*x)/32 - 3*b**3*d**2*e**3*x**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 +
1)*asinh(c + d*x)**2/16 - 3*b**3*d**2*e**3*x**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/128 - 9*b**3*d*e**3*x**2*
asinh(c + d*x)/32 + 9*b**3*e**3*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**2/32 + 45*b**3*e**3*x*s
qrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/256 - 3*b**3*e**3*asinh(c + d*x)**3/(32*d) - 45*b**3*e**3*asinh(c + d*x)/(
256*d), Ne(d, 0)), (c**3*e**3*x*(a + b*asinh(c))**3, True))

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)^3*(a+b*arcsinh(d*x+c))^3,x, algorithm="giac")

[Out]

integrate((d*e*x + c*e)^3*(b*arcsinh(d*x + c) + a)^3, x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (c\,e+d\,e\,x\right )}^3\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*e + d*e*x)^3*(a + b*asinh(c + d*x))^3,x)

[Out]

int((c*e + d*e*x)^3*(a + b*asinh(c + d*x))^3, x)

________________________________________________________________________________________